General Graph Question

Hello all,
I am interested in learning about graphs and shortest paths in Java, particularly for problems like the Word Ladder from SIGCSE 2001. I have read up about graphs in a couple Algorithm / Java books, but they mostly describe the structures and behaviors of graphs and algorithms to traverse graphs (BFS, DFS, etc), but I'm having a difficult time understanding how to "construct" a graph.
For example, if one was interested in finding the shortest path between two five letter words in a graph (start vertex, end vertex), you could read in all five letter words that are to be tested into an unordered Array. After the words are all stored in the unordered array, you could BFS the unordered array which would take the starting vertex (a word from the unordered array), add it to a queue, mark that vertex as visited, remove it from the queue, and then find the next word that only has a one letter difference than the last, add that to the queue, mark it as visited, search for the next word that's only one letter away from the second in the unordered array, etc?
money <----- Starting vertex
honey
etc
etc
etc
(ending vertex)
I don't know if I did a good job of explaining what I'm asking... Generally what it boils down to is that I'm having a difficult time understanding the actual concept of graphs and how they are constructed, assuming you have a bunch of five letter words (Word Ladder) and you need to find the shortest path between first word and the last word you supply.
I've tried searching around the net for a good example of graphs actually being constructed and searched through in Java, but something isn't clicking with me. Is there anybody that could assist in helping me understand this concept?
Any help would be appreciated, and I apologize in advance if I'm not supposed to post general questions in this forum.

Building the graph is the lesser of the tedious jobs you have to do: every word represents a node. An edge from node A to node B occurs if word A differs in only one letter with word B. If you represent the graph as nodes having an outgoing edge list then B contains A in its list only if A contains B in its list, i.e. the graph isn't directed. e.g. "honey" is one 'hop' away from "money" because "money" is one 'hop' away from "honey"; the 'hop' is the edge. Once you have that complete graph you can play with shortest path algorithms etc.
kind regards,
Jos
ps. personally I'd build a BitSet per node indicating which other node(s) are one hop away from that node. The words can be simply stored in a linear array.

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